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When drawing ray diagrams for concave mirrors, I was advised to:

  1. draw a ray that is parallel to the principle axis, the reflected ray will pass through the focus.

  2. Draw a ray through the center (C), the reflected ray will be same as the incident ray but opposite in direction.

I understand the reasoning behind point 2. as the incident angle will equal to zero. However, I don't understand why every single incoming parallel ray will pass through this certain point called the focus. Why not choose any other point. Is there any proof as to why this happen or was this just experimentally found?

Main question: why do parallel rays pass through the focus?

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    $\begingroup$ The two rules are a reasonable approximation of what happens when light parallel to the mirror axis shines on a parabolic mirror. This can be found both experimentally, as well as mathematically, if one applies the laws of reflection to a parabolic surface. Unless I am mistaken, you are having some doubts about the universality of these rules? The good, and bad news is, that you are right. Reflection on real mirrors is much more complicated than that, and leads to many optical errors in devices like telescopes. Having said that, this approximation is still useful for many simple problems. $\endgroup$ – CuriousOne Aug 20 '14 at 5:35
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You've probably heard of a curve called the parabola, and you probably interpret this as meaning it's a function something like $y = x^2$. However there is another way to define the parabola. If you draw a line (called the directrix) and then choose a point (called the focus) not on that line then the set of points that are an equal distance from the directrix and the focus form a parabola.

Parabola

(the picture is from this web site)

All conic sections have a focus. For the circle the focus is the centre, and you may have heard that the planets orbit in ellipses with the Sun at one focus. Anyhow, with some effort you can prove that any light ray from the focus reflects off the parabola parallel to the axis of symmetry, or any ray parallel to the axis of symmetry is reflected through the focus. This property is the basis of parabolic reflectors.

But what has this to do with spherical mirrors? Well, as long as you keep the curvature of a spherical mirror small it is very similar to a parabola and it shares the property of reflecting parallel rays onto (almost) a single point. Hence statement (1) in your question.

However because a spherical mirror isn't the same as a parabolic mirror the approximation breaks down as the spherical mirror gets bigger. Spherical mirrors actually focus parallel rays onto a surface called a caustic, not onto a single point.

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